arXiv subjects

Sam van der Poel

Publications and source records attributed to Sam van der Poel.

4 recordsLinked to original sources

arXiv2026

Excluding an induced star in dense random graphs

For fixed $k\geq3$, we study the asymptotic number and typical structure of dense graphs with no induced copy of the star $K_{1,k}$. We solve the associated graphon variational problems both at fixed constant edge density $\gamma$ and for the conditioned Erd\H{o}s--R\'enyi random graph $G(n,p)$ for constant $p$. As consequences, we obtain explicit formulas for the entropy density of induced-$K_{1,k}$-free graphs with $\Theta(n^2)$ edges and for the large deviation rate function for the event that $G(n,p)$ is induced-$K_{1,k}$-free. The entropy density exhibits a second-order phase transition at an explicit critical density $\gamma_k$, while the rate function exhibits a first-order phase transition at a critical parameter $p_k$. We completely characterize the optimizers of both variational problems. Both models have parameter values for which there are infinitely many optimal graphons, but there is always a unique graphon that represents the typical structure in cut metric. We refine the graphon-level results by giving a detailed structural description of both models. For supercritical parameters, each random graph model is the complement of a $(k-1)$-partite graph with high probability. In the subcritical regime of the fixed-density model, the typical structure is the disjoint union of the complement of a $(k-1)$-partite graph, and a sparse remainder. In the subcritical regime of the conditioned Erd\H{o}s--R\'enyi random graph, a typical sample has $o(n^2)$ edges.

math.CO↗

arXiv2026

Spectral recovery of a planted triangle-dense subgraph

Given a simple graph on $n$ vertices and a parameter $k$, the triangle-densest-$k$-subgraph problem is known to be computationally hard in the worst case. To circumvent the computational hardness, we study an average-case model where a triangle-dense subgraph on $k$ vertices is planted in an Erd\H{o}s-R\'enyi random graph on $n$ vertices. For the recovery of the planted subgraph, we propose a simple spectral algorithm and a semidefinite program, both of which use a graph matrix whose entries are local signed triangle counts. Theoretical guarantees for these algorithms are established through spectral analysis of the graph matrix. Finally, we provide evidence showing a statistical-to-computational gap analogous to that for the planted clique problem. The computational threshold in terms of the subgraph size $k$ is at least $\sqrt{n}$ in the framework of low-degree polynomial algorithms, while the information-theoretic threshold is at most logarithmic in $n$.

cs.DS↗

arXiv2025

The typical structure of dense claw-free graphs

We analyze the asymptotic number and typical structure of claw-free graphs at constant edge densities. The first of our main results is a formula for the asymptotics of the logarithm of the number of claw-free graphs of edge density $γ\in (0,1)$. We show that the problem exhibits a second-order phase transition at edge density $γ^\ast=\frac{5-\sqrt{5}}{4}$. The asymptotic formula arises by solving a variational problem over graphons. For $γ\geqγ^\ast$ there is a unique optimal graphon, while for $γ<γ^\ast$ there is an infinite set of optimal graphons. By analyzing more detailed structure, we prove that for $γ<γ^\ast$, there is in fact a unique graphon $W$ such that almost all claw-free graphs at edge density $γ$ are close in cut metric to $W$. We also analyze the probability of claw-freeness in the Erdős-Rényi random graph $G(n,p)$ for constant $p$, obtaining a formula for the large-deviation rate function for claw-freeness. In this case, the problem exhibits a first-order phase transition at $p^\ast=\frac{3-\sqrt{5}}{2}$, separating distinct structural regimes. At the critical point $p^\ast$, the corresponding graphon variational problem has infinitely many solutions, and we again pinpoint a unique optimal graphon that describes the typical structure of $G(n,p^\ast)$ conditioned on being claw-free.

math.CO↗

arXiv2024

MLRegTest: A Benchmark for the Machine Learning of Regular Languages

Synthetic datasets constructed from formal languages allow fine-grained examination of the learning and generalization capabilities of machine learning systems for sequence classification. This article presents a new benchmark for machine learning systems on sequence classification called MLRegTest, which contains training, development, and test sets from 1,800 regular languages. Different kinds of formal languages represent different kinds of long-distance dependencies, and correctly identifying long-distance dependencies in sequences is a known challenge for ML systems to generalize successfully. MLRegTest organizes its languages according to their logical complexity (monadic second order, first order, propositional, or monomial expressions) and the kind of logical literals (string, tier-string, subsequence, or combinations thereof). The logical complexity and choice of literal provides a systematic way to understand different kinds of long-distance dependencies in regular languages, and therefore to understand the capacities of different ML systems to learn such long-distance dependencies. Finally, the performance of different neural networks (simple RNN, LSTM, GRU, transformer) on MLRegTest is examined. The main conclusion is that performance depends significantly on the kind of test set, the class of language, and the neural network architecture.

cs.LG↗